Child migrants arriving at Dover may be turned away by Kent County Council (KCC) within days as its services are overwhelmed, the authority has said.
KCC has taken its first steps in legal action against Home Secretary Priti Patel and wants her to make other councils take "their fair share".
It comes days after Ms Patel lost a High Court legal fight over the Napier Barracks asylum centre in Folkestone.
The Home Office said it continued to encourage more areas to do their part.
Bridget Chapman, of Kent Refugee Action Network, said local authorities across the country would be willing to take child asylum-seekers into care if the government provided greater funding.
"Councils aren't going to accept the responsibility without it being funded properly," she said.
"These young people are incredible," she said. "With the right foundation they are going to contribute an enormous amount to our communities."
Conservative council leader Roger Gough said it was "a repeat of the same crisis of nine months ago", when services for unaccompanied asylum-seeking children reached breaking point.
With more young people arriving this year compared to last, he said the pressures were "severe".
He told the BBC: "We are now in a situation which is not sustainable, and we do not wish to... suspend our legal duties, but that is very close to where we are," he added.
So far this year, 242 lone child migrants have arrived on Kent shores and been passed to children's services, but only 52 have been moved to other local authorities under a voluntary transfer scheme, KCC said.
It said the authority is now caring for 403 unaccompanied minors - nearly double the number the government said it was safe to have.
Mr Gough wants the existing voluntary transfer scheme to be compulsory, because "those others who do not participate frankly put a lot of pressure those who do".
A proposed judicial review claims there has been a "refusal and/or continuing failure by the Home Secretary to exercise her powers to prepare a mandatory scheme".
Mr Gough said: "We have not seen what is most needed: a robust National Transfer Scheme that prevents port authorities such as Kent coming under unmanageable pressure."
He said Kent would no longer be able to safely accept new arrivals before the end of the week.
From that point, Border Force will be asked to place youngsters directly into other local authorities around the country from Dover port.
Roger Gale, the Conservative MP for North Thanet, said: "Kent basically is now full. We cannot safely take any more, so the Home Secretary has got to do something about it."
He said Ms Patel faced two questions. "Why has nothing been done over the last nine months? And what is she going to do about it?" he asked.
Claim 'to be issued'
KCC has served a formal Letter Before Action to the Home Office.
The council said without any substantive response to its proposals by 17 June, it would issue a claim for judicial review.
A spokeswoman for the Home Office said: "We recognise the longstanding role that Kent County Council has played in supporting unaccompanied asylum-seeking children and are extremely grateful for their contribution.
"We continue to encourage more areas to join the National Transfer Scheme and do their part.
"We have already consulted on how to improve the scheme to make it fairer, the outcome of which will be published very shortly."
While there was a rise in young migrants entering KCC's care last year, the total number of unaccompanied children seeking asylum across the UK fell by more than 1,300.
From April 2020 to March 2021, 2,044 asylum claims were made by lone children, compared to 3,530 in the previous 12 months. The majority of asylum-seeking children arrive with their families, Home Office figures show.
The number of people reaching the UK in small boats rose in 2020 due to a reduction in flights and ferries during the pandemic, but the "overall organised immigration crime threat reduced", the National Crime Agency has said.
"According to the Home Office, numbers are down on last year, but the method of arrival has changed and it's just more visible," Ms Chapman said.